Classification of Solutions

Videos, examples, solutions to help Grade 8 students know the conditions for which a linear equation will have a unique solution, no solution, or infinitely many solutions.

New York State Common Core Math Grade 8, Module 4, Lesson 7

Common Core Math Grade 8, Module 4, Lesson 7 Worksheets (pdf)

Lesson 7 Outcome

• Students know the conditions for which a linear equation will have a unique solution, no solution, or infinitely many solutions.

Lesson 7 Summary

• There are three classifications of solutions to linear equations: one solution (unique solution), no solution, or infinitely many solutions.

Equations with no solution will, after being simplified, have coefficients of x that are the same on both sides of the equal sign and constants that are different. For example, x + b = x + c, where b, c are constants that are not equal. A numeric example is 8x + 5 = 8x - 3.

Equations with infinitely many solutions will, after being simplified, have coefficients of x and constants that are the same on both sides of the equal sign. For example, x + a = x + a, where a is a constant. A numeric example is 6x + 1 = 1 + 6x.

NYS Math Module 4 Grade 8 Lesson 7 Classwork

Exercises 1–3
Solve each of the following equations for x.

1. 7x - 3 = 5x + 5
   
2. 7x - 3 = 7x + 5
   
3. 7x - 3 = -3 + 7x
   Note: if the coefficients of x are different and the value of the constants are the same, the only solution is x = 0. For example, 2x + 12 = x + 12

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Exercises 1–3

Activity: What can we see in an equation that will tell us about the solution to the equation?

Exercises 4–10
Give a brief explanation as to what kind of solution(s) you expect the following linear equations to have. Transform the equation into a simpler form if necessary.

4. 11x - 2x + 15 = 8 + 7 + 9x
   
   
5. 3(x -14) + 1 = -4x + 5
   
   
6. -3x + 32 - 7x = -2(5x + 10)
   
   
7. 1/2(8x + 26) = 13 + 4x
   
   
8. Write two equations that have no solutions.
   
   
9. Write two equations that have one unique solution each.
   
   
10. Write two equations that have infinitely many solutions.

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